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<h3 style="text-align: center;">Approach to equilibrium: Molecular dynamics</h3>

<p class="header_title">Introduction</p>

<p>The program implements a molecular dynamics simulation of a
Lennard-Jones fluid
in two dimensions. The particles are initialized in the middle third of
the simulation cell. The number of particles in each of the three cells
is plotted.</p>

<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.approach.LJgasApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p class="header_title">Algorithm</p>

<p>The algorithm for simulating the evolution of the model can be
summarized by the following steps:</p>

<ol>

<li>The particles are placed at random in the center of the box with the constraint that no
two particles can be closer than the length &#963; = 1 in the Lennard-Jones potential. This constraint
prevents the initial force between any two particles from being too big,
which would lead to the breakdown of the numerical method used to solve the
differential equations. The
velocity of each particle is assigned at random and then the velocity of the
center of mass is set to zero.</li>

<li>The total force on each particle is computed and the positions and velocities are updated according to the Verlet algorithm.</li>

<li> The time is increased by the time step &#916;t.</li>

</ol>

<p class="header_title">Problems</p>
<ol>

<li>Choose N = 27, n<sub>1</sub> = 0, n<sub>2</sub> = N, and n<sub>3</sub> = 0. What
is the qualitative behavior of
n<sub>1</sub>, n<sub>2</sub>, and n<sub>3</sub>, the number of particles in each third of
the box, as a function of the time t? Does the system
appear to show a direction of time? Choose various values of N
that are multiples of three up to
N = 270. Is the direction of time better defined for larger
N?</li>

<li>Suppose that we made a video of the motion of the particles
considered in Problem 1. Would
you be able to tell if the video were played forward or backward for the
various values of N? 
Would you be willing to make an even bet about the direction
of time? Does your conclusion about the direction of
time become more certain as N increases?</li>

<li>How long
does it take before the system appears to be in equilibrium? What is
your criterion for equilibrium?</li>

<li>From watching the motion of the particles, describe the
nature of the boundary conditions that are used in the
simulation.</li>

<li>*Run the system with N = 270 until t = 5. Then press the <tt>Reverse</tt> button, which reverse all the velocities, and continue the simulation. Does the system return to its initial state? Repeat for other times. Is there a time at which the system does not return to its initial state? Repeat for N = 27. What can you conclude about the N dependence of reversibility?</li>

</ol>

<p class="header_title">Java Classes Used</p>
<ul>

<li>LJgas</li>

<li>LJgasApp</li>
</ul>

<p class="small">Updated 27 February 2007.</p>
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